Beamforming in MIMO systems is a well known technique that improves the performance of a wireless link using multiple antennas at the transmitter and receiver, and prior knowledge of the channel characteristic. In MIMO systems, the input symbols are mixed across the transmit antennas such that the MIMO channel signal level observed by the receiver is improved. The modulation method Orthogonal Frequency Division Multiplexing (OFDM) simplifies wideband beamforming by separating a frequency selective wideband channel into multiple subcarriers, each subcarrier occupying a narrowband spectrum of the wideband channel. Each subcarrier can be beamformed in the frequency domain using narrowband beamforming techniques known in the prior-art. Beamforming is currently supported by OFDM based IEEE wireless local area network (WLAN) standards such as IEEE WLAN 802.11n and 802.16e.
FIG. 1 shows a wireless MIMO link with two input spatial streams S1, S2 104 applied to MIMO transmitter 102 which combines them in a separable manner, and couples them to transmit output antennas 106, 108. MIMO receiver 114 receives the combined streams on antennas 110 and 112 and processes the received signals to decode two receive streams R1, R2, 116 which carry the same information as original streams S1 and S2 104. The transmit antennas 106 and 108 carry transmit streams X1 and X2, respectively, and the receive antennas 110 and 112 receive combined signals Y1 and Y2, respectively. The generalized linear equations of operation for FIG. 1 are:Y=HWS+N 
where:
Y is the receive matrix,
H is the channel characterization matrix,
W is the transmit beamforming matrix,
S is a matrix contains the streams of data to be transmitted, and
N is system noise.
The following equations describe the matrix form of the 2×2 MIMO system of FIG. 1:
      [                                        y            ⁢                                                  ⁢            1                                                            y            ⁢                                                  ⁢            2                                ]    =                    [                                                            h                ⁢                                                                  ⁢                11                                                                    h                ⁢                                                                  ⁢                12                                                                                        h                ⁢                                                                  ⁢                21                                                                    h                ⁢                                                                  ⁢                22                                                    ]            *              [                                                            w                ⁢                                                                  ⁢                11                                                                    w                ⁢                                                                  ⁢                12                                                                                        w                ⁢                                                                  ⁢                21                                                                    w                ⁢                                                                  ⁢                22                                                    ]            *              [                                                            s                ⁢                                                                  ⁢                1                                                                                        s                ⁢                                                                  ⁢                2                                                    ]              +          [                                                  n              ⁢                                                          ⁢              1                                                                          n              ⁢                                                          ⁢              2                                          ]      
FIG. 2 shows a transmitter and receiver link across a wireless channel for 3×3 MIMO with two spatial input streams S1, S2 202, which are mixed and transmitted over three antennas. The associated transmitted signals x1, x2, and x3 are modified by a beamforming matrix W 212 which is preceded and followed by signal processing 214 and 210, respectively, further modified by the communication channel which is represented by the channel matrix H, and arrive at the receiver 206 antennas Y1, Y2, Y3. Each of the receiver antennas gathers a sum of modified signals from each of the transmitted antennas, which can be represented as:
      [                                        y            ⁢                                                  ⁢            1                                                            y            ⁢                                                  ⁢            2                                                            y            ⁢                                                  ⁢            3                                ]    =                    [                                                            h                ⁢                                                                  ⁢                11                                                                    h                ⁢                                                                  ⁢                12                                                                    h                ⁢                                                                  ⁢                13                                                                                        h                ⁢                                                                  ⁢                21                                                                    h                ⁢                                                                  ⁢                22                                                                    h                ⁢                                                                  ⁢                23                                                                                        h                ⁢                                                                  ⁢                31                                                                    h                ⁢                                                                  ⁢                32                                                                    h                ⁢                                                                  ⁢                33                                                    ]            *              [                                                            w                ⁢                                                                  ⁢                11                                                                    w                ⁢                                                                  ⁢                12                                                                                        w                ⁢                                                                  ⁢                21                                                                    w                ⁢                                                                  ⁢                22                                                                                        w                ⁢                                                                  ⁢                31                                                                    w                ⁢                                                                  ⁢                32                                                    ]            *              [                                                            s                ⁢                                                                  ⁢                1                                                                                        s                ⁢                                                                  ⁢                2                                                    ]              +          [                                                  n              ⁢                                                          ⁢              1                                                                          n              ⁢                                                          ⁢              2                                                                          n              ⁢                                                          ⁢              3                                          ]      
where:
yk is the received signal at the kth receive path
hik is the channel between the ith transmit antenna and the kth receive antenna
nk is the noise signal at the kth receive path
s1, s2 are the two transmitted symbols
wi, j are the beamforming coefficients and W is the beamforming (mixing) matrix
The value transmitted on each antenna x can be represented as
      [                                        x            ⁢                                                  ⁢            1                                                            x            ⁢                                                  ⁢            2                                                            x            ⁢                                                  ⁢            3                                ]    =            [                                                  w              ⁢                                                          ⁢              11                                                          w              ⁢                                                          ⁢              12                                                                          w              ⁢                                                          ⁢              21                                                          w              ⁢                                                          ⁢              22                                                                          w              ⁢                                                          ⁢              31                                                          w              ⁢                                                          ⁢              32                                          ]        *          [                                                  s              ⁢                                                          ⁢              1                                                                          s              ⁢                                                          ⁢              2                                          ]      
By computing the values of W using knowledge of the channel characteristic H, it is possible to improve the receive signal quality at the receiver.